Actuarial Mathematics (MA310)

Graham Ellis http://hamilton.nuigalway.ie

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Section B: Project Appraisal

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Methods that can be used to decide between alternative investment projects: NPV and Accumulated Prot Internal Rate of Return Payback period Discounted Payback Period Measures of investment return: Money Weighted Rate of Return Time Weighted Rate of Return Limited Internal Rate of Return

The Risk Discount Rate is important to the appraisal.

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

The Risk Discount Rate is important to the appraisal.

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

The Risk Discount Rate is important to the appraisal.

NPVB (i)

NPVA(i) 16.8 i

In general: NPV (i1 ) > 0 project protable @ROI i1

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Example A project has an initial outlay on 1/5/2006 of 4,000,000 . Three months later a further expenditure of 2,000,000 will be required. On 1/10/2007 income will be received of 20,000 a month payable in arrears for 25 years. The income increases by 5% per annum compound on 1st October each year; the rst increase of 5% occurs on 1st October 2010. Calculate the net present value of the project at a rate of 5% per annum.

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

v = 1/1.05 NPV =< 4000 > + < 2000 > v 12 + (12)(20)v 1 12 a3|5 3 5

(12)

12 +(12)(20)v 3 12 (1.05v + (1.05v )2 + + (1.05v )22 )a1|

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

v = 1/1.05 NPV =< 4000 > + < 2000 > v 12 + (12)(20)v 1 12 a3|5 3 5

(12)

12 +(12)(20)v 3 12 (1.05v + (1.05v )2 + + (1.05v )22 )a1|

NPV =< 998, 800 >

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Internal Rate of Return

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Internal Rate of Return IRR = i such that NPV (i ) = 0 (IRR also called Yield To Redemption)

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Internal Rate of Return IRR = i such that NPV (i ) = 0 (IRR also called Yield To Redemption) Software company example:

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Example An investor is considering whether to invest in either/both of the following loans. Loan X For a purchase price of 10,000 the investor will receive 1,000 pa payable quaterly in arrears for 15| . Loan Y For a purchase price of 11,000 the investor will receive an income of 605 pa, payable annually in arrears for 18|, and a return of his outlay at the end of this period. The investor may borrow or lend money at 4% pa . Would you advise the investor to invest in either loan? If so, which would be the more protable?

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

A company is to choose between D and E, both of which would be nanced by a loan, repayable only at the end of the project. The company must pay 6.25% pa on money borrowed, but can earn only 4% on money invested in its deposit account.

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Discounted Payback Period

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Discounted Payback Period = Number of years before project gets out of the red

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Discounted Payback Period = Number of years before project gets out of the red DPP is the smallest value of t such that A(t) 0 wheret

A(t) =s<t

cs (1 + j1 )ts +0

(s)(1 + j1 )ts ds

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Discounted Payback Period = Number of years before project gets out of the red DPP is the smallest value of t such that A(t) 0 wheret

A(t) =s<t

cs (1 + j1 )ts +0

(s)(1 + j1 )ts ds

Note: A(t) depends only on j1 .

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Discounted Payback Period = Number of years before project gets out of the red DPP is the smallest value of t such that A(t) 0 wheret

A(t) =s<t

cs (1 + j1 )ts +0

(s)(1 + j1 )ts ds

Note: A(t) depends only on j1 . If DPP=t1 then

T

A(T ) = A(t1 )(1+j2 )T t1 +

t>t1

ct (1+j2 )T t +t1

(t)(1+j2 )T t dt

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

DPP assumption: A(t) changes sign only once. Appraisal: A project with a shorter DPP is preferable to one with longer DPP because it will produce prots earlier. Payback Period = DPP with j1 = 0. Do NOT use Payback Period.

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Example The business plan for a new company that has obtained a 5| lease for operating a local bus service is shown below. Cashow Timing Amount (000s) Initial set up costs @start < 100 > Advertizing for income 1 month +200 Purchase of vehicles 3 months < 2000 > Passenger fares Continuous from 3 months +1000 pa Sta costs Continuous from 3 months < 400 > pa Resale value of assets 5 years +500 Calculate DPP for the project assuming it will be nanced by a exible loan facility based on an eective ROI of 10% pa .

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

A(t) =< 100 > (1.1)t + 200(1.1)t 12 2000(1.1)t 12

t

3 12

(1000 400)(1.1)ts ds3

A(t) =< 1899.2 > (1.1)t 12 + 600s 10% 3

t 12 |

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

A(t) =< 100 > (1.1)t + 200(1.1)t 12 2000(1.1)t 12

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Example A company is considering investing in one of a number of projects. The chosen project is to be nanced by a bank loan of 500,000 at an eective rate of interest of 9% per annum. Any surplus funds may be invested at 6% per annum eective. One of the projects has an outlay of 500,000 and income of 70,000 per annum payable continuously for 20 years. (a) Calculate the discount payback period for the project. (b) Calculate the accumulated prot after 20 years.

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Yield 199% 20%

Prot @ 1| 199 2,000

If 10,000 readily available then B gives a higher prot. If 9,999 must be borrowed then all depends on the ROI of borrowings. e.g. if borrowing ROI=18% then Prot of B = 200. This is greater than the prot on A but maybe prefer A as its less hassle.

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Conclusion: There is more to consider in arriving at a decision that just a comparison of prot.

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Cashow Are CF requirements for the project consistent with the businesss other needs. (e.g. Could it need future borrowing?) Over what period will the prot emerge, & how will prot be used? Is project worth carrying out if potential prot is small?

Graham Ellis http://hamilton.nuigalway.ie

Actuarial Mathematics (MA310)

Borrowing Can the necessary funds be borrowed when required? Are time limits or other restrictions imposed on borrowing? What ROI? Any security over other assets?